2 Answers
2

I forgive you would be for something sincere and serious, like a hurtful insult or even serious crimes.

For something formal or written, you can say I accept your apology.

A very common phrase, as @CookieMonster mentions, is It's OK or It's all right.

Sometimes, people apologize out of politeness, just in case the other person might be offended. In such a situation, people often respond with No worries or No problem. For example, if you texted your friend that you were running ten minutes late to meet her, hopefully she would say, "No worries!"

Another very common phrase is No need to apologize.

There are other more colorful expressions, such as:

Don't worry about it.

Forget about it!

Don't even stress!

Don't even trip!

If you don't forgive someone, you can usually add "not" or "don't" to the phrases above:

"it's all right" is not the correct spelling, as far as I know.
– Michael RybkinNov 18 '17 at 0:57

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I don't think "alright" is a real word, at least not in the U.S.
– RingoNov 18 '17 at 0:59

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I always thought that "alright" is an adjective that means "fine". "all right" literally means "100% correct" or "each one in a group is correct".
– Michael RybkinNov 18 '17 at 1:02

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"Alright" is a variant spelling of "all right", in all of the uses of that expression. It's less formal (and some would argue incorrect), but it's commonplace and growing in usage, although still less common than "all right".
– PMVNov 18 '17 at 1:17

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Seems like everything I learned as a student is just being thrown out the window. :-( The rule I learned was that it's "all right" and definitely NOT "alright" -- that just because a lot of people misspell it doesn't make it OK to misspell it. Now that sounds old-fashioned, but it makes a lot of sense to me on a certain level.
– RingoNov 18 '17 at 1:43

In Canada, it's acceptable to respond back to "Sorry" with "Sorry," especially if one person has bumped into the other person on the street or something like that. This is distinctly Canadian though and other English speakers sometimes make fun of Canadians for overusing the word sorry.